Background: Social networks are often highly skewed, meaning that the vast majority of the population has only few contacts whereas a small minority has a large number of contacts. These highly connected individuals may play an important role in case of an infectious disease outbreak. Methods: We propose a novel strategy of finding and immunizing highly connected individuals and evaluate this strategy by computer simulations, using a stochastic, individual-and network-based simulation approach. A small random sample of the population is asked to list their acquaintances, and those who are mentioned most frequently are offered vaccination. This intervention is combined with case isolation and contact tracing. Results: Asking only 10% of the population for 10 acquaintances each and vaccinating the most frequently named people strongly diminishes the magnitude of an outbreak which would otherwise have exhausted the available isolation units and gone out of control. It is extremely important to immunize all identified highly connected individuals. Omitting a few of them because of unsuccessful vaccination jeopardizes the overall success, unless non-immunized individuals are taken under surveillance. Conclusions: The strategy proposed in this paper is particularly successful because it attacks the very point from which the transmission network draws its strength: the highly connected individuals. Current preparedness and containment plans for smallpox and other infectious diseases may benefit from such knowledge.